Mathematical Exploration and Discovery at Scale (terrytao.wordpress.com)

Bogdan Georgiev, Javier Gómez‑Serrano, Adam Zsolt Wagner and coauthor released an arXiv paper reporting a large-scale experimental study (67 problems) of AlphaEvolve, a DeepMind tool that uses an LLM to evolve short pieces of code rather than evolving numeric inputs directly. The LLM generates, mutates and recombines programmatic candidates whose outputs are evaluated by a scoring function; prompts can include PDFs or hints, and the LLM can even propose discretization and hyperparameter choices. The authors provide a repository with prompts and evolution traces and compare AlphaEvolve’s behavior across analysis, combinatorics and geometry problems, showing it often matches or improves on expert-tuned optimizers (e.g., packing and calculus-of-variations tasks) and sometimes rediscovers exact analytic optimizers (Talenti functions). Small improvements (e.g., for sum–difference exponents) even inspired subsequent rigorous theory. Technically significant are three practical advantages and caveats: (1) scale and reuse—verification/prompts for one problem adapt to many variants; (2) robustness and interpretability—solutions are often inspectable code rather than opaque vectors; (3) stochastic LLM mutations provide exploration that can escape local extrema. Important limitations include susceptibility to verifier “exploits” (necessitating exact/interval arithmetic and conservative scoring), bias toward known literature solutions when training data overlaps, one-sided bounds on conjectures, and weaker performance on problems like analytic number theory. Overall the paper demonstrates LLM-driven programmatic optimization is a powerful, pragmatic tool for automated mathematical exploration while underscoring verification and domain-limit caveats.